bifurcation analysis of a chemostat model of plasmid bearing and plasmid free competition with pulsed input

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 343719, pages http://dx.doi.org/10.1155/2014/343719 Research Article Bifurcation Analysis of a Chemostat Model of Plasmid-Bearing and Plasmid-Free Competition with Pulsed Input Zhong Zhao, Baozhen Wang, Liuyong Pang, and Ying Chen Department of Mathematics, Huanghuai University, Zhumadian, Henan 463000, China Correspondence should be addressed to Zhong Zhao; zhaozhong8899@163.com Received 18 April 2014; Revised 17 May 2014; Accepted 24 May 2014; Published 15 June 2014 Academic Editor: Mohamad Alwash Copyright © 2014 Zhong Zhao et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited A chemostat model of plasmid-bearing and plasmid-free competition with pulsed input is proposed The invasion threshold of the plasmid-bearing and plasmid-free organisms is obtained according to the stability of the boundary periodic solution By use of standard techniques of bifurcation theory, the periodic oscillations in substrate, plasmid-bearing, and plasmid-free organisms are shown when some conditions are satisfied Our results can be applied to control bioreactor aimed at producing commercial producers through genetically altered organisms Introduction Biofermentation has become an active area of research on the continuous cultivation of microorganism in recent years [1– 3] The chemostat is an important laboratory apparatus used to continuously culturing microorganisms [2–8] It can be used to investigate microbial growth because the parameters are measurable, the experiments are reasonable, and the mathematics is tractable [9] Fermentations using genetically modified (recombinant) microorganisms typically contain two kinds of cellsrecombinant cells and wild-type cells The former contains a genetically inserted plasmid (a foreign DNA molecule that can exist independent of the host chromosome and can replicate autonomously) which is responsible for the coding functions that result in the synthesis of a desired protein Wild-type or plasmid-free cells not contain this plasmid and therefore cannot generate the protein Nevertheless, they consume nutrients, grow, and multiply From this perspective, plasmid-free cells may thus be considered undesirable, and different methods are employed to check their proliferation As recombinant (or plasmid-bearing) cells have to support a larger metabolic load than plasmid-free cells, their growth rates are smaller In addition, these cells lose their plasmids during the fermentation process With the scientific technology, the importance of the genetically altered technology is widely recognized Therefore, it is necessary to understand the dynamic behavior of the fermentation process Xiang and Song [10] analyze a simple chemostat model for plasmid-bearing and plasmid-free organisms with the pulsed substrate and linear functional response They prove that system is permanent if the impulsive period is less than some critical value Shi et al [11] consider a new Monod type chemostat model with delayed growth response and pulsed input in the polluted environment Normally, the velocity of the enzyme reaction increases with the increase in substrate concentration Some enzymes, however, display the phenomenon of excess substrate inhibition, which means that large amounts of substrate can have the adverse effect and actually slow the reaction down The Monod function does not account for any inhibitory effect at high substrate concentration Therefore, it is crucial to choose a response function showing the excess substrate inhibition Pal et al [12] introduce the MonodHaldene functional responses into a three-tier model of phytoplankton, zooplankton, and nutrient in order to investigate the phenomenon of excess substrate inhibition Therefore, we introduce the Monod-Haldene functional Journal of Applied Mathematics response into the following chemostat model with pulsed input: In the absence of the plasmid-bearing organism, system (2) is reduced to 𝜇1 𝑥1 𝑆 𝜇2 𝑥2 𝑆 𝑆 ̇ (𝑡) = −𝑄𝑆 − − , 𝛿 (𝐴 + 𝑆 + 𝐵𝑆 ) 𝛿 (𝐴 + 𝑆 + 𝐵𝑆2 ) 𝑥1̇ (𝑡) = 𝑥1 ( 𝑥̇ (𝑡) = −𝑥 − 𝜇1 (1 − 𝑞) 𝑆 − 𝑄) , 𝐴 + 𝑆 + 𝐵𝑆2 𝜇2 𝑆 𝑞𝜇1 𝑆𝑥1 − 𝑄) + , 𝑥2̇ (𝑡) = 𝑥2 ( 𝐴 + 𝑆 + 𝐵𝑆 𝐴 + 𝑆 + 𝐵𝑆2 𝑛𝜏 𝑡 ≠ , 𝑄 Δ𝑆 = 𝑆0 𝜏, Δ𝑥1 = 0, Δ𝑥2 = 0, 𝑡= 𝑥̇ (𝑡) = −𝑥 − 𝑦̇ (𝑡) = 𝑦 ( (1) 𝑡 ≠ 𝑛𝜏, (3) Δ𝑥 = 𝜏, 𝑡 = 𝑛𝜏 Δ𝑧 = 0, This nonlinear system has a simple periodic solution For our purpose, we present the solution in this section We add the first and the second equations of (3) We have 𝑛𝜏 , 𝑄 𝑑 (𝑥 + 𝑧) = − (𝑥 + 𝑧) 𝑑𝑡 𝑚1 𝑥𝑦 𝑚2 𝑥𝑧 − , 𝑎1 + 𝑥 + 𝑎2 𝑥 𝑎1 + 𝑥 + 𝑎2 𝑥2 𝑚1 (1 − 𝑞) 𝑥 − 1) , 𝑎1 + 𝑥 + 𝑎2 𝑥2 (4) Taking the variable change 𝑠 = 𝑥 + 𝑦, the system (3) can be rewritten as 𝑑𝑠 = −𝑠, 𝑑𝑡 𝑡 ≠ 𝑛𝜏, + 𝑠 (𝑛𝜏 ) = 𝑠 (𝑛𝜏) + 𝜏, (5) 𝑡 = 𝑛𝜏 Thus, we have the following Lemma System (5) has a positive periodic solution 𝑠̃(𝑡) and for any solution 𝑠(𝑡) of system (5) |𝑠(𝑡) − 𝑠̃(𝑡)| → as 𝑡 → ∞, where 𝑠̃(𝑡) = (𝜏 exp(−(𝑡−𝑛𝜏))/(1−exp(−𝜏))), 𝑡 ∈ (𝑛𝜏, (𝑛+1)𝜏], and 𝑠̃(0+ ) = 𝜏/(1 − 𝑒−𝜏 ) Proof Clearly, 𝑠̃(𝑡) is a positive periodic solution of the system (5) Any solution 𝑠(𝑡) of system (5) is 𝑠(𝑡) = (𝑠(0) − (𝜏/(1 − 𝑒−𝜏 )))𝑒−𝑡 +̃𝑠(𝑡), 𝑡 ∈ (𝑛𝜏, (𝑛+1)𝜏], 𝑛 ∈ 𝑍+ Hence, |𝑠(𝑡)−̃𝑠(𝑡)| → as 𝑡 → ∞ Lemma Let (𝑥(𝑡), 𝑧(𝑡)) be any solution of system (5) with initial condition 𝑥(0) ≥ 0, 𝑧(0) > 0, and then lim𝑡 → ∞ |𝑥(𝑡) + 𝑧(𝑡) − 𝑠̃(𝑡)| = Lemma (5) shows that the periodic solution 𝑠̃(𝑡) is uniquely invariant manifold of system (3) Therefore, one has < 𝑥(𝑡) ≤ 𝑀, < 𝑧(𝑡) ≤ 𝑀, where 𝑀 = 𝜏/(1 − 𝑒−𝑇 ) Theorem For system (3), one has the following 𝑚1 𝑞𝑥𝑦 𝑚2 𝑥 − 1) + , 𝑎1 + 𝑥 + 𝑎2 𝑥2 𝑎1 + 𝑥 + 𝑎2 𝑥2 𝑡 ≠ 𝑛𝜏, Δ𝑥 = 𝜏, Δ𝑦 = 0, Δ𝑧 = 0, 𝑚2 𝑥𝑧 , 𝑎1 + 𝑥 + 𝑎2 𝑥2 𝑚2 𝑥 𝑧̇ (𝑡) = 𝑧 ( − 1) , 𝑎1 + 𝑥 + 𝑎2 𝑥2 where Δ𝑆 = 𝑆(𝑛𝜏+ /𝑄) − 𝑆(𝑛𝜏/𝑄), Δ𝑥1 = 𝑥1 (𝑛𝜏+ /𝑄) − 𝑥1 (𝑛𝜏/𝑄), Δ𝑥2 = 𝑥2 (𝑛𝜏+ /𝑄) − 𝑥2 (𝑛𝜏/𝑄), and 𝑛 ∈ 𝑍+ , 𝑍+ = {1, 2, } 𝐴 (𝐴 > 0) can be interpreted as the halfsaturation constant in the absence of any inhibitory effect 𝐵 (𝐵 > 0) is the measure of inhibitory effect 𝑆(𝑡) denotes the nutrient concentration at time 𝑡, and 𝑥1 (𝑡) denotes the concentration of plasmid-bearing organism at time 𝑡 𝑥2 (𝑡) is the concentration of plasmid-free organism at time 𝑡 𝜇1 and 𝜇2 are the uptake constant of the microorganism 𝛿 is the constant yield (It is reasonable to assume that the yield constants for two organisms are the same since they are the same organism just with or without the plasmid) 𝑆0 represents the input concentration of the nutrient each time, and the probability that a plasmid is lost in reproduction is represented by 𝑞 (0 < 𝑞 < 1) 𝑄 (0 < 𝑄 < 1) is the washout proportion of the chemostat each time 𝑛𝜏/𝑄 is the period of the pulse The variables in the above system may be rescaled by measuring 𝑥(𝑡) ≡ 𝑆(𝑡)/𝑆0 , 𝑦(𝑡) ≡ 𝑥1 (𝑡)/𝛿𝑆0 , 𝑧(𝑡) ≡ 𝑥2 (𝑡)/𝑆0 𝛿, and 𝑡 = 𝑄𝑡, and then we have the following system: 𝑧̇ (𝑡) = 𝑧 ( The Behavior of the Substrate and Plasmid-Free Organism Subsystem 𝑡 = 𝑛𝜏, where 𝑚1 = 𝜇1 /𝑄, 𝑚2 = 𝜇2 /𝑄, 𝑎1 = 𝐴/𝑆0 , 𝑎2 = 𝐵𝑆0 (2) 𝜏 (1) If (1/𝜏) ∫0 (𝑚2 𝑠̃(𝑙)/(𝑎1 + 𝑠̃(𝑙) + 𝑎2 𝑠̃(𝑙)2 ))𝑑𝑙 < 1, system (3) has a uniquely globally stable boundary 𝜏-periodic solution (𝑥𝑒 (𝑡), 𝑧𝑒 (𝑡)), where 𝑥𝑒 (𝑡) = 𝑠̃, 𝑧𝑒 (𝑡) = 𝜏 (2) If (1/𝜏) ∫0 (𝑚2 𝑠̃(𝑙)/(𝑎1 +̃𝑠(𝑙)+𝑎2 𝑠̃(𝑙)2 ))𝑑𝑙 > 1, system (3) has a globally asymptotically stable positive 𝜏-periodic solution (𝑥𝑠 (𝑡), 𝑧𝑠 (𝑡)), and one has 𝜏 𝑚2 (̃𝑠 (𝑡) − 𝑧𝑠 (𝑡)) 𝑎1 + 𝑠̃ (𝑡) − 𝑧𝑠 (𝑡) + 𝑎2 (̃𝑠 (𝑡) − 𝑧𝑠 (𝑡)) ∫ 𝑑𝑡 = (6) Journal of Applied Mathematics 𝜏 Proof (1) If (1/𝜏) ∫0 (𝑚2 𝑠̃(𝑙)/(𝑎1 + 𝑠̃(𝑙) + 𝑎2 𝑠̃(𝑙)2 ))𝑑𝑙 < 1, it is obvious that 𝑚2 𝑠̃ (𝑙) 𝜏 𝑑𝑙 − 1) 𝑡) 𝑧 (𝑡) ≤ 𝑧 (0) exp (( ∫ 𝜏 𝑎1 + 𝑠̃ (𝑙) + 𝑎2 𝑠̃(𝑙)2 𝑡 𝑧𝑠 (𝑡) = 𝑧 (𝑡, 𝑧0∗ ) , (7) × exp (∫ 𝑝1 (𝑙) 𝑑𝑙) , where 𝑝1 (𝑡) = (𝑚2 𝑠̃(𝑙)/(𝑎1 + 𝑠̃(𝑙) + 𝑎2 𝑠̃(𝑙) )) − 𝜏 (1/𝜏) ∫0 (𝑚2 𝑠̃(𝑙)/(𝑎1 + 𝑠̃(𝑙) + 𝑎2 𝑠̃(𝑙)2 ))𝑑𝑙 𝑝1 (𝑡) is 𝜏-periodic 𝜏 piecewise continuous function in view of (1/𝜏) ∫0 𝑝1 (𝑙)𝑑𝑙 = 𝜏 For (1/𝜏) ∫0 (𝑚2 𝑠̃(𝑙)/(𝑎1 + 𝑠̃(𝑙) + 𝑎2 𝑠̃(𝑙)2 ))𝑑𝑙 − < 0, we obtain 𝑧(𝑡) that tends exponentially to zero as 𝑡 → +∞ From Lemma and system (5), we have lim𝑡 → ∞ |𝑥(𝑡) − 𝑠̃(𝑡)| = 𝜏 (2) If (1/𝜏) ∫0 (𝑚2 𝑠̃(𝑙)/(𝑎1 + 𝑠̃(𝑙) + 𝑎2 𝑠̃(𝑙)2 ))𝑑𝑙 > 1, we consider system (3) in its stable invariant manifold 𝑠̃(𝑡); that is, 𝑚2 (̃𝑠 (𝑡) − 𝑧 (𝑡)) 𝑧 (𝑡) 𝑑𝑧 − 𝑧, = 𝑑𝑡 𝑎1 + 𝑠̃ (𝑡) − 𝑧 (𝑡) + 𝑎2 (̃𝑠 (𝑡) − 𝑧 (𝑡))2 (8) Suppose 𝑧(𝑡, 𝑧0 ) is a solution of (8) with initial condition 𝑧0 ∈ (0, 𝑠̃(0)], and we obtain 𝑧 (𝑡, 𝑧0 ) 𝑛𝜏 𝑚2 (̃𝑠 (𝑡) − 𝑧 (𝑡)) 𝑎1 + 𝑠̃ (𝑡) − 𝑧 (𝑡) + 𝑎2 (̃𝑠 (𝑡) − 𝑧 (𝑡))2 𝜏 𝑚2 (̃𝑠 (𝑡) − 𝑧𝑠 (𝑡)) 𝑎1 + 𝑠̃ (𝑡) − 𝑧𝑠 (𝑡) + 𝑎2 (̃𝑠 (𝑡) − 𝑧𝑠 (𝑡)) 𝑡 ∈ (𝑛𝜏, (𝑛 + 1) 𝜏 ] For (9), we have the following properties: (i) the function 𝐹(𝑧0 ) = 𝑧(𝑡, 𝑧0 ), 𝑧0 ∈ (0, 𝑠̃(0)] is an increasing function; (ii) < 𝑧(𝑡, 𝑧0 )𝑧 < 𝑠̃(0), 𝑡 ∈ [0, ∞) is a continuous function; (iii) 𝑧(𝑡, 0) = 0, 𝑡 ∈ [0, ∞) is a solution 𝑧0 𝑚2 (̃𝑠 (𝑙) − 𝑧 (𝑙)) − 1) 𝑑𝑙) 𝑎1 + 𝑠̃ (𝑙) − 𝑧 (𝑙) + 𝑎2 (̃𝑠 (𝑙) − 𝑧 (𝑙))2 (10) 𝜏 (12) 𝑡 𝑚2 (̃𝑠 (𝑡) − 𝑧𝑠 (𝑡)) 𝑎1 + 𝑠̃ (𝑡) − 𝑧𝑠 (𝑡) + 𝑎2 (̃𝑠 (𝑡) − 𝑧𝑠 (𝑡)) 𝐹 (𝑧 (𝑡, 𝑧0 )) = ∫ 𝑑𝑡 − 𝑡 (13) 𝐹 (𝑧 (𝜏, 𝑧0 )) = ln ( 𝑧 (𝜏, 𝑧0 ) ), 𝑧0 𝑧0 ∈ (0, 𝑠̃ (0) ] , (14) and it is obvious that 𝐹(𝑧(𝑛𝜏), 𝑧0∗ ) = For any 𝑧0 ∈ (0, 𝑠̃(0)], by the theorem on the differentiability of the solutions on the initial values, 𝜕𝑧(𝑡, 𝑧0 )/𝜕𝑧0 exists Furthermore, 𝜕𝑧(𝑡, 𝑧0 )/𝜕𝑧0 holds for 𝑡 ∈ (0, ∞) (otherwise, there exists < 𝑧1 < 𝑧2 < 𝑠̃(0) such that 𝑧(𝑡0 , 𝑧1 ) = 𝑧(𝑡0 , 𝑧2 ) for 𝑡1 > 0, which is a contradiction to the different flows of system (8) not to intersect, and we have 𝑠̃(𝑙) > 𝑧(𝑙, 𝑧0 ) for 𝑙 ∈ [0, 𝜏] So, we obtain that (15) Therefore, 𝐹(𝑧(𝜏, 𝑧0 )) is monotonously decreasing continuous function for 𝑧0 ∈ [0, 𝑠̃(0)] Now, we choose 𝜀 such as < 𝜀 < 𝑧0∗ < 𝑠̃(0), and we have the following cases: ln 𝑧 (𝜏, 𝑧0 ) − ln 𝑧0 < 0, if 𝑧0∗ < 𝑧0 < 𝑠̃ (0) , if 𝑧0∗ = 𝑧0 , ln 𝑧 (𝜏, 𝑧0 ) − ln 𝑧0 = 0, ln 𝑧 (𝜏, 𝑧0 ) − ln 𝑧0 > 0, (16) if 𝜀 < 𝑧0 < 𝑧0∗ Furthermore, we obtain the following equations: The periodic solution of (9) satisfies the following equation: = 𝑧0 exp (∫ ( 𝑑𝑡 = 1, 𝑑𝐹 (𝑧 (𝜏, 𝑧0 )) < 𝑑𝑧0 (9) 𝜏 where 𝑧0∗ = 𝑧𝑠 (0) In order to prove the stability of the periodic solution 𝑧𝑠 (𝑡), we define a function 𝐹 : (𝑡, 𝑧0 ) → 𝑅 ∈ [0, ∞) × [0, 𝑠̃(0)] as the following: −1) 𝑑𝑙) , 𝑧 (𝑛𝜏) = 𝑧0 , (11) Noticing (8), we have ≤ 𝑧0 ≤ 𝑠̃ (0) 𝑡 𝑥𝑠 (𝑡) = 𝑠̃ (𝑡) − 𝑧 (𝑡, 𝑧0∗ ) From (9), we obtain ∫ = 𝑧 (𝑛𝜏) exp (∫ ( If 𝑚2 > 𝑚2∗ , system (8) has a uniquely positive periodic solution We denote the positive periodic solution And we denote 𝑚2∗ = (𝜏/ ∫0 (̃𝑠(𝑙)/(𝑎1 + 𝑠̃(𝑙) + 𝑎2 𝑠̃(𝑙)2 ))𝑑𝑙) By (i) and (ii), system (8) has a stable solution 𝑧𝑒 (𝑡) = for 𝑚2 < 𝑚2∗ By Lemma 2, we have lim𝑡 → ∞ |𝑥(𝑡) − 𝑠̃(𝑡)| = 𝑧0 > 𝑧 (𝜏, 𝑧0 ) > ⋅ ⋅ ⋅ 𝑧 (𝑛𝜏, 𝑧0 ) > 𝑧0∗ , 𝑧0 < 𝑧 (𝜏, 𝑧0 ) < ⋅ ⋅ ⋅ 𝑧 (𝑛𝜏, 𝑧0 ) < 𝑧0∗ , if 𝑧0∗ < 𝑧0 ≤ 𝑠̃ (0) , if 𝜀 < 𝑧0 < 𝑧0∗ (17) Suppose that lim 𝑧 (𝑛𝜏, 𝑧0 ) = 𝑎 𝑛→∞ (18) We will prove that the solution 𝑧(𝑡, 𝑎) is 𝜏-periodic Because the function 𝑧𝑛 (𝑡) = 𝑧(𝑡 + 𝑛𝜏, 𝑧0 ) is also a solution of system (8) and 𝑧𝑛 (0) → 𝑎 as 𝑛 → ∞, we have that 𝑧(𝜏, 𝑎) = lim𝑛 → ∞ 𝑧𝑛 (𝜏) = 𝑎 by the continuous dependence of Journal of Applied Mathematics the solutions on the initial values Hence, the solution 𝑧(𝑡, 𝑎) is 𝜏-periodic The periodic solution 𝑧(𝑡, 𝑧0∗ ) is unique and 𝑎 = 𝑧0∗ Let 𝜀 > be given, by the theorem on the continuous dependence of the solutions on the initial values, there exists a 𝛿 > such that 󵄨󵄨 ∗ 󵄨 󵄨󵄨𝑧 (𝑡, 𝑧0 ) − 𝑧 (𝑡, 𝑧0 )󵄨󵄨󵄨 < 𝜀, (19) if |𝑧0 − 𝑧0∗ | < 𝛿 and ≤ 𝑡 ≤ 𝜏 Choose 𝑛1 > such as |𝑧(𝑛𝜏, 𝑧0 ) − 𝑧0∗ | < 𝛿 for 𝑛 > 𝑛1 Then, |𝑧(𝑡, 𝑧0 ) − 𝑧(𝑡, 𝑧0∗ )| < 𝜀 for 𝑡 > 𝑛𝜏, which shows that 󵄨 󵄨 lim 󵄨󵄨𝑧 (𝑡, 𝑧0 ) − 𝑧 (𝑡, 𝑧0∗ )󵄨󵄨󵄨 = 0, 𝑧0 ∈ (0, 𝑠̃ (0) ] (20) 𝑛→∞ 󵄨 For convenience, we suppose 𝑚2 > 𝑚2∗ and denote that 𝑚1∗ = 𝜏 𝑚2 𝑥𝑠 (𝑙) (𝑎1 − 𝑎2 𝑥𝑠2 (𝑙)) (𝑎1 + 𝑥𝑠 (𝑙) + 𝑎2 𝑥𝑠2 (𝑙)) 𝑑𝑙) < (21) This conclusion will be used in Section 3 The Bifurcation of the System In order to investigate the properties of system (2), we add the first, second, and third equations of system (2) and take variable change 𝑠 = 𝑥 + 𝑦 + 𝑧, and the following lemma is obvious (1 − 𝑞) ∫0 (𝑧𝑠 (𝑙) / (𝑎1 + 𝑧 (𝑙) + 𝑎2 𝑧𝑠2 (𝑙))) 𝑑𝑙 󵄨 󵄨 lim 󵄨󵄨𝑥 (𝑡) + 𝑦 (𝑡) + 𝑧 (𝑡) − 𝑠̃ (𝑡)󵄨󵄨󵄨 = −1 − ( 𝑑Φ (𝑡) ( =( ( 𝑑𝑡 ( ( (22) 𝑚2 (𝑎1 − 𝑎2 𝑥𝑠 ) 𝑧𝑠 (𝑎1 + 𝑥𝑠 + 𝑎2 𝑥𝑠2 ) 𝑚2 (𝑎1 − 𝑎2 𝑥𝑠 ) 𝑧𝑠 (𝑎1 + 𝑥𝑠 + 𝑎2 𝑥𝑠2 ) − (2) If 𝑚2 > 𝑚2∗ and 𝑚1 < 𝑚1∗ , then system (2) has a unique globally asymptotically stable boundary 𝜏-periodic solution (𝑥𝑠 (𝑡)), 0, 𝑧𝑠 (𝑡) (3) If 𝑚2 > 𝑚2∗ and 𝑚1 > 𝑚1∗ , then boundary 𝜏-periodic solution (𝑥𝑠 (𝑡)), 0, 𝑧𝑠 (𝑡) of system (2) is unstable Proof The proof of (1) is easy; we want to prove (2) and (3) The local stability of periodic solution (𝑥𝑠 (𝑡), 0, 𝑧𝑠 (𝑡)) may be determined by considering the behavior of small amplitude of the solution Define 𝑥 (𝑡) = 𝑤 (𝑡) + 𝑥𝑠 (𝑡) , 𝑦 (𝑡) = V (𝑡) , 𝑧 (𝑡) = 𝑤 (𝑡) + 𝑧𝑠 (𝑡) (24) There may be written 𝑢 (0) 𝑢 (𝑡) ( V (𝑡) ) = Φ (𝑡) ( V (0) ) , 𝑤 (0) 𝑤 (𝑡) ≤ 𝑡 ≤ 𝜏, (25) where Φ(𝑡) = {𝜑𝑖𝑗 }𝑖,𝑗=1,2,3 satisfies 𝑚1 𝑥𝑠 𝑎1 + 𝑥𝑠 + 𝑎2 𝑥𝑠2 −1 + (23) (1) If 𝑚2 < 𝑚2∗ , then system (2) has a unique globally asymptotically stable boundary 𝜏-periodic solution (̃𝑠(𝑡), 0, 0) Lemma Let 𝑋(𝑡) = (𝑥(𝑡), 𝑦(𝑡), 𝑧(𝑡)) be any solution of system (2) with 𝑋(0) > 0, and then 𝑡→∞ 󵄨 Theorem Let (𝑥(𝑡), 𝑦(𝑡)𝑧(𝑡)) be any solution of system (2) with 𝑋(0) > 0, and we obtain the following For system (3), we obtain lim𝑡 → ∞ |𝑥−𝑥𝑠 | = and lim𝑡 → ∞ |𝑧− 𝑧𝑠 | = for any solution (𝑥(𝑡), 𝑧(𝑡)) with the initial condition 𝑥(0) ≥ 0, 𝑧(0) > From the 𝜏-periodic solution 𝑧𝑠 being asymptotically stable, we obtain the multiplier 𝜇 of 𝑧𝑠 , which satisfies 𝜇 = exp (∫ 𝜏 𝜏 𝑚2 𝑥𝑠 𝑎1 + 𝑥𝑠 + 𝑎2 𝑥𝑠2 − 𝑚2 𝑥𝑠 𝑎1 + 𝑥𝑠 + 𝑎2 𝑥𝑠2 𝑚1 𝑞𝑥𝑠 𝑎1 + 𝑥𝑠 + 𝑎2 𝑥𝑠2 −1 + ) ) ) Φ (𝑡) , ) ) (26) 𝑚1 (1 − 𝑞) 𝑥𝑠 𝑎1 + 𝑥𝑠 + 𝑎2 𝑥𝑠2 ) and Φ(0) = 𝐼 is the identity matrix Hence, the fundament solution matrix is ∗ 𝜑11 (𝜏) 𝜑12 (𝜏) ∗∗ 𝜑21 (𝜏) 𝜑22 (𝜏) ), Φ (𝜏) = ( 𝜏 𝑚1 (1 − 𝑞) 𝑥𝑠 (𝑙) 0 exp (∫ ( − 1) 𝑑𝑙) 𝑎1 + 𝑥𝑠 (𝑙) + 𝑎2 𝑥𝑠 (𝑙) (27) Journal of Applied Mathematics and there is no need to calculate the exact form (∗) and ∗∗ as it is not required in the analysis that follows = The eigenvalues of the matrix Φ(𝜏) are 𝜇3 𝜏 exp(∫0 ((𝑚1 (1 − 𝑞)𝑥𝑠 (𝑙)/(𝑎1 + 𝑥𝑠 (𝑙) + 𝑎2 𝑥𝑠2 (𝑙))) − 1)𝑑𝑙), 𝜇1 , and 𝜇2 , where 𝜇1 and 𝜇2 are determined by the following matrix: Φ (𝜏) = ( 𝜑11 (𝜏) 𝜑12 (𝜏) ) 𝜑21 (𝜏) 𝜑22 (𝜏) (28) 𝜇1 and 𝜇2 are also the multipliers of the local linearization system of (3) According to Theorem 3, we have that 𝜇1 < and 𝜇2 < 𝜏 If 𝑚2 > 𝑚2∗ and 𝑚1 < 𝑚1∗ and 𝜇3 = exp(∫0 ((𝑚1 (1 − 𝑞)𝑥𝑠 (𝑙)/(𝑎1 + 𝑥𝑠 (𝑙) + 𝑎2 𝑥𝑠2 (𝑙))) − 1)𝑑𝑙) < 1, the boundary periodic solution (𝑥𝑠 (𝑡), 0, 𝑧𝑠 (𝑡)) of system (2) is locally 𝑡 asymptotically stable In view of 𝑦(𝑡) ≤ 𝑦(0) exp(∫0 ((𝑚1 (1 − 𝑞)𝑥𝑠 (𝑙)/(𝑎1 + 𝑥𝑠 (𝑙) + 𝑎2 𝑥𝑠2 (𝑙))) − 1)𝑑𝑙), we obtain 𝑦(𝑡) → as 𝑡 → ∞ From lim𝑡 → ∞ |𝑥(𝑡) + 𝑦(𝑡) + 𝑧(𝑡) − 𝑠̃(𝑡)| = 0, we have lim𝑡 → ∞ |𝑥(𝑡) + 𝑧(𝑡) − 𝑠̃(𝑡)| = Now, using Theorem 3, we have lim𝑡 → ∞ |𝑧(𝑡) − 𝑧̃𝑠 (𝑡)| = and lim𝑡 → ∞ |𝑥(𝑡) − 𝑥̃𝑠 (𝑡)| = 𝜏 If 𝑚2 > 𝑚2∗ , 𝑚1 > 𝑚1∗ and 𝜇3 = exp(∫0 ((𝑚1 (1 − 𝑞)𝑥𝑠 /(𝑎1 + 𝑥𝑠 + 𝑎2 𝑥𝑠2 )) − 1)𝑑𝑙) > 1, the boundary periodic solution (𝑥𝑠 (𝑡), 0, 𝑠𝑠 (𝑡)) of system (2) is unstable The proof is completed Let 𝐵 denote the Banach space of continuous, 𝜏-periodic function 𝑁 : [0, 𝜏] → 𝑅2 In the set 𝐵, we introduce the norm |𝑁|0 = sup0≤𝑡≤𝜏 |𝑁(𝑡)| with which 𝐵 becomes a Banach space with the uniform convergence topology For convenience, just like [13], we introduce the following Lemmas and Lemma Suppose 𝑎𝑖𝑗 ∈ 𝐵 𝜏 𝜏 (a) If ∫0 𝑎22 (𝑠)𝑑𝑠 ≠ 0, ∫0 𝑎11 (𝑠)𝑑𝑠 ≠ 0, then the linear homogenous system 𝑑𝑦1 = 𝑎11 𝑦1 + 𝑎12 𝑦2 , 𝑑𝑡 𝑑𝑦2 = 𝑎22 𝑦1 , 𝑑𝑡 𝑑𝑦2 = 𝑎22 𝑦1 + 𝑓2 , 𝑑𝑡 Ł (𝑓1 , 𝑓2 ) ≡ (𝐿 (𝑎12 𝐿 𝑓2 ) , 𝐿 𝑓2 ) 𝜏 (29) (30) has, for every (𝑓1 , 𝑓2 ) ∈ 𝐵 × 𝐵, a unique solution (𝑥1 , 𝑥2 ) ∈ 𝐵 × 𝐵 and the operator 𝐿 : 𝐵×𝐵 → 𝐵×𝐵 defined by (𝑥1 , 𝑥2 ) = 𝐿(𝑓1 , 𝑓2 ) is linear and compact If we define that 𝑥2󸀠 = 𝑎22 𝑥2 +𝑓2 has a unique solution 𝑥2 ∈ 𝐵 and the operator 𝐿 : 𝐵 → 𝐵 defined by 𝑥2 = 𝐿 𝑓2 is linear and compact Further, 𝑥1󸀠 = 𝜏 𝑎11 𝑥1 +𝑓3 (𝑓3 ∈ 𝐵) has a unique solution (since ∫0 𝑎11 (𝑠)𝑑𝑠 ≠ 0) (31) 𝜏 (b) If ∫0 𝑎22 (𝑠)𝑑𝑠 = 0, ∫0 𝑎11 (𝑠)𝑑𝑠 ≠ 0, then (29) has exactly one independent solution in 𝐵 × 𝐵 𝜏 Lemma Suppose 𝑎 ∈ 𝐵 and (1/𝜏) ∫0 𝑎(𝑙)𝑑𝑙 = Then, 𝑥1󸀠 = 𝑎𝑥 + 𝑓(𝑓 ∈ 𝐵) has a solution 𝑥 ∈ 𝐵 if and only if 𝜏 𝑙 (1/𝜏) ∫0 𝑓(𝑙)(exp(− ∫0 𝑎(𝑠)𝑑𝑠)𝑑𝑙) = By Lemma 7, in its invariant manifold 𝑠̃ = 𝑥(𝑡)+𝑦(𝑡)+𝑧(𝑡), system (2) is reduced to a equivalently nonautonomous system as the following: 𝑚 𝑥 (̃𝑠 − 𝑥 − 𝑦) 𝑚1 𝑥𝑦 − , 𝑎1 + 𝑥 + 𝑎2 𝑥2 𝑎1 + 𝑥 + 𝑎2 𝑥2 𝑡 ≠ 𝑛𝜏, 𝑚 (1 − 𝑞) 𝑥 − 1) , 𝑦̇ (𝑡) = 𝑦 ( 𝑎1 + 𝑥 + 𝑎2 𝑥2 𝑥̇ (𝑡) = −𝑥 − Δ𝑥 = 𝜏, 𝑡 = 𝑛𝜏, Δ𝑦 = 0, (32) and if 𝑚2 > 𝑚2∗ , the boundary periodic solution (𝑥𝑠 , 0) is locally asymptotically stable provided with 𝑚1 < 𝑚1∗ and it is unstable provided with 𝑚1 > 𝑚1∗ ; hence 𝑚1∗ practices as a bifurcation threshold For system (32), we have the following result Theorem If 𝑚2 > 𝑚2∗ and 𝑎1 − 2𝑎2 𝑥𝑠2 ≥ hold, then there exists a constant 𝜆 , such that 𝑚1 ∈ (𝑚1∗ , 𝑚1∗ + 𝜆 ), and there exists a solution (𝑥, 𝑦) satisfying < 𝑥 < 𝑥𝑠 , 𝑦 > 0, 𝑧 = 𝑠̃ − 𝑥 − 𝑦 > Hence, system (2) has a positive 𝜏-periodic solution (𝑥, 𝑦, 𝑠̃ − 𝑥 − 𝑦) Proof Let 𝑥1 = 𝑥 − 𝑥𝑠 , 𝑥2 = 𝑦, and then system (32) becomes 𝑑𝑥1 = 𝐹11 (𝑥𝑠 , 𝑠̃) 𝑥1 + 𝑥2 + 𝑔1 (𝑥1 , 𝑥2 ) , 𝑑𝑡 𝑑𝑥2 = 𝐹22 (𝑚1 , 𝑥𝑠 ) 𝑥2 + 𝑔2 (𝑥1 , 𝑥2 ) , 𝑑𝑡 has no nontrivial solution in 𝐵 × 𝐵 In this case, the nonhomogeneous system 𝑑𝑦1 = 𝑎11 𝑦1 + 𝑎12 𝑦2 + 𝑓1 , 𝑑𝑡 in 𝐵 and 𝑥1 = 𝐿 𝑓3 define a linear, compact operator 𝐿 : 𝐵 → 𝐵 Then, we have (33) where 𝑚2 (̃𝑠 − 𝑥𝑠 ) 𝑚2 𝑥𝑠 (𝑎1 − 2𝑎2 𝑥𝑠 ) 𝐹11 (𝑥𝑠 , 𝑠̃) = − 𝑎1 − 2𝑎2 𝑥𝑠2 (𝑎1 + 𝑥𝑠 + 𝑎2 𝑥𝑠2 ) −1− 𝐹12 (𝑚1 , 𝑥𝑠 , 𝑠̃) = 𝑚2 𝑠̃𝑥𝑠 (1 + 2𝑎2 𝑥𝑠 ) (𝑎1 + 𝑥𝑠 + 𝑎2 𝑥𝑠2 ) 𝑚2 𝑥𝑠 (𝑎1 − 2𝑎2 𝑥𝑠2 ) (𝑎1 + 𝑥𝑠 + − 𝑎2 𝑥𝑠2 ) 𝑚1 𝑥𝑠 (𝑎1 + 𝑥𝑠 + 𝑎2 𝑥𝑠2 ) 𝐹22 (𝑚1 , 𝑥𝑠 ) = + , 𝑚2 𝑠̃𝑥𝑠 (1 + 2𝑎2 𝑥𝑠 ) (𝑎1 + 𝑥𝑠 + 𝑎2 𝑥𝑠2 ) , (1 − 𝑞) 𝑚1 𝑥𝑠 (𝑎1 + 𝑥𝑠 + 𝑎2 𝑥𝑠2 ) − (34) Journal of Applied Mathematics 6e − 13 1.4 5e − 13 1.2 x 4e − 13 y 3e − 13 2e − 13 0.8 1e − 13 120 160 140 180 t 200 220 120 240 140 160 (a) 180 t 200 220 240 (b) 4e − 05 3e − 05 0.8 z 2e − 05 z 1e − 05 0.6 0.4 0.2 0 120 140 160 180 t 200 220 240 (c) 0.4 x 0.8 1.2 0.2 0.4 0.6 y 0.8 (d) Figure 1: Time series of the system (2) with pulse 𝑎1 = 0.2, 𝑎2 = 0.2, 𝑚1 = 1.3, 𝑞 = 0.16, and 𝑇 = 0.5 𝜏 We know that ∫0 (((1 − 𝑞)𝑚1 𝑥𝑠 (𝑙)/(𝑎1 + 𝑥𝑠 (𝑙) + 𝑎2 𝑥𝑠2 )2 (𝑙))) − 1)𝑑𝑙 ≠ 0, and, by Lemma 7, we can equivalently write system (33) as the operator equation If 𝑦1 × 𝑦2 ∈ 𝐵 × 𝐵 is a solution of (37) for some 𝑚1 > 0, then (𝑦1 , 𝑦2 ) satisfies (𝑥1 , 𝑥2 ) = 𝐿∗ (𝑥1 , 𝑥2 ) + 𝐺 (𝑥1 , 𝑥2 ) , 𝑑𝑦1 = 𝐹11 (𝑥𝑠 , 𝑠̃) 𝑦1 + 𝐹12 (𝑥𝑠 , 𝑠̃) 𝑦2 , 𝑑𝑡 (35) where 𝐺 (𝑥1 , 𝑥2 ) = 𝐿 (𝐹12 (𝑥𝑠 , 𝑠̃) 𝑔2 (𝑥1 , 𝑥2 ) + 𝑔1 (𝑥1 , 𝑥2 ) , 𝐿 𝑔2 (𝑥1 , 𝑥2 )) (36) ∗ Here, 𝐿 : 𝐵 × 𝐵 → 𝐵 × 𝐵 is linear and compact 𝐺 : 𝐵 × 𝐵 → 𝐵 × 𝐵 is continuous and compact (since 𝐿 and 𝐿 are compact) and satisfies 𝐺 = 𝑜(|(𝑥1 , 𝑥2 )|0 ) near (0, 0) A nontrivial solution (𝑥1 , 𝑥2 ) ≠ (0, 0) for some 𝑚1 > yields a solution (𝑥, 𝑦) = (𝑥𝑠 + 𝑥1 , 𝑥2 ) ≠ (0, 0) of system (32) Thus, the existence of the periodic solution of system (2) can be considered as the bifurcation problem of system (35) Now, we apply local bifurcation technique to (35) As is well known, bifurcation can occur only at the nontrivial solution of linearized problem (𝑦1 , 𝑦2 ) = 𝐿∗ (𝑦1 , 𝑦2 ) , 𝑚1 > (37) 𝑑𝑦2 = 𝐹22 (𝑥𝑠 ) 𝑦2 𝑑𝑡 (38) By virtue of Lemma (b) and system (38), system (37) has one nontrivial solution in 𝐵 × 𝐵 if and only if 𝑚1∗ = 𝑚1 Therefore, we obtain one piecewise continuous periodic solution of system (35), which is all nontrivial solutions expect for (𝑚1∗ , 0, 0) Now, we investigate the solution (𝑚1 , 𝑥1 , 𝑥2 ) near the bifurcation point (𝑚1∗ , 0, 0) by expanding 𝑚1 and 𝑥1 , 𝑥2 in a Lyapunov-Schmidt series: 𝑚1 = 𝑚1∗ + 𝜆𝜀 + ⋅ ⋅ ⋅ , 𝑥1 = 𝑥11 𝜀1 𝑥12 𝜀2 + ⋅ ⋅ ⋅ , 𝑥2 = 𝑥21 𝜀 + 𝑥22 𝜀2 + ⋅ ⋅ ⋅ , (39) Journal of Applied Mathematics 7e − 17 6e − 17 5e − 17 0.8 4e − 17 y x 0.6 3e − 17 2e − 17 0.4 1e − 17 120 140 160 180 t 200 220 120 240 140 160 (a) 180 t 200 220 240 (b) 0.398 0.396 0.8 z 0.394 0.6 z 0.392 0.4 0.2 0.39 120 140 160 180 t 200 220 240 0.4 x 0.8 1.2 0.2 (c) 0.6 y 0.4 0.8 (d) Figure 2: Time series of the system (2) with pulse 𝑎1 = 0.2, 𝑎2 = 0.2, 𝑚1 = 1.3, 𝑚2 = 1.5, 𝑞 = 0.2, and 𝑇 = 0.8 where 𝑥𝑖𝑗 ∈ 𝐵, 𝜀 is a small parameter If we substitute those series into the system (32) and equate coefficients of 𝜀 and 𝜀2 , we find that 󸀠 = 𝐹11 (𝑥𝑠 , 𝑠̃) 𝑥11 + 𝐹12 (𝑚1∗ , 𝑥𝑠 , 𝑠̃) 𝑥21 , 𝑥11 󸀠 = 𝐹22 (𝑚1∗ , 𝑥𝑠 ) 𝑥21 , 𝑥21 𝜏 𝑥11 < for all 𝑡 (since 𝑚2 > 𝑚2∗ and (14), ∫0 ((𝑚2 (̃𝑠 − 𝑥𝑠 )/(𝑎1 − 2𝑎2 𝑥𝑠2 ))−(𝑚2 𝑥𝑠 (𝑎1 −2𝑎2 𝑥𝑠2 )/(𝑎1 +𝑥𝑠 +𝑎2 𝑥𝑠2 )2 ) – 1−(𝑚2 𝑠̃𝑥𝑠 (1+ 𝜏 2𝑎2 𝑥𝑠 )/(𝑎1 + 𝑥𝑠 + 𝑎2 𝑥𝑠2 )2 ))𝑑𝑙 = − ∫0 ((𝑚2 𝑥𝑠 (𝑎1 − 2𝑎2 𝑥𝑠2 )/(𝑎1 + 𝑥𝑠 + 𝑎2 𝑥𝑠2 )2 ) + (𝑚2 𝑠̃𝑥𝑠 (1 + 2𝑎2 𝑥𝑠 )/(𝑎1 + 𝑥𝑠 + 𝑎2 𝑥𝑠2 )2 ))𝑑𝑙 < implies that Green’s function for the first equation in (38) is positive) From Lemma 7, we obtain that 󸀠 𝑥12 = 𝐹11 (𝑥𝑠 , 𝑠̃) 𝑥12 + 𝐹12 (𝑚1∗ , 𝑥𝑠 , 𝑠̃) 𝑥22 , 󸀠 = 𝐹22 (𝑚1∗ , 𝑥𝑠 ) 𝑥22 + 𝑥22 × (𝜆 + 𝑥21 (1 − 𝑞) 𝑎1 + 𝑥𝑠 + 𝑎2 𝑥𝑠2 (40) (𝑎1 − 2𝑎2 𝑥𝑠2 ) ) 𝑎1 + 𝑥𝑠 + 𝑎2 𝑥𝑠2 𝜏 𝑚1∗ 𝑥11 (𝑡) 𝑥21 (𝑡) (𝑎1 − 𝑎2 𝑥𝑠2 (𝑡)) (𝑎1 + 𝑥𝑠 (𝑡) + 𝑎2 𝑥𝑠2 (𝑡)) 𝜆 = − ((∫ 𝑚1∗ 𝑥11 Thus, (𝑥1 , 𝑥2 ) ∈ 𝐵 × 𝐵 must exist a solution of (37) Choose the specific solution such as the initial condition 𝑥21 (0) = 1, and then we have 𝑡 𝑚1∗ 𝑥𝑠 (𝑙) 𝑥21 = exp (∫ ((1 − 𝑞) − 1) 𝑑𝑙) > 0, 𝑎1 + 𝑥𝑠 (𝑙) + 𝑥𝑠2 (𝑙) (41) 𝑡 × exp (∫ ( (1 − 𝑞) 𝑚1∗ 𝑥𝑠 (𝑙) − 1) 𝑎1 + 𝑥𝑠 (𝑙) + 𝑎2 + 𝑥𝑠2 (𝑙) 𝑑𝑙) 𝑑𝑡) × (∫ 𝜏 𝑥21 (𝑡) 𝑎1 + 𝑥𝑠 (𝑡) + 𝑎2 𝑥𝑠2 (𝑡) Journal of Applied Mathematics 0.104 0.8 x 0.103 0.6 y 0.102 0.101 0.4 0.1 0.2 120 140 160 180 t 200 220 240 120 140 160 (a) 180 t 200 220 240 (b) 0.416 0.414 0.412 0.4 0.41 z 0.35 0.408 0.3 0.406 0.404 z 0.25 0.2 0.402 0.15 0.4 0.1 0.398 0.2 120 140 160 180 t 200 220 240 (c) 0.4 x 0.6 0.8 1.2 0.04 0.06 0.08 y 0.1 (d) Figure 3: Time series of the system (2) with pulse 𝑎1 = 0.2, 𝑎2 = 0.2, 𝑚1 = 2, 𝑚2 = 1.5, 𝑞 = 0.2, and 𝑇 = 0.8 𝑡 × exp (∫ ( (1 − 𝑞) 𝑚1∗ 𝑥𝑠 (𝑙) − 1) 𝑎1 + 𝑥𝑠 (𝑙) + 𝑎2 + 𝑥𝑠2 (𝑙) −1 𝑑𝑙) 𝑑𝑡) ) > 0, (42) provided with 𝑎1 − 2𝑎2 𝑥𝑠2 ≥ 0, which shows that near the bifurcation point (𝑚1∗ , 0, 0), there exists a constant 𝜆 , such that 𝑚1 ∈ (𝑚1∗ , 𝑚1∗ + 𝜆 ) Thus, system (29) has a solution (𝑥, 𝑦) ∈ (𝐵 × 𝐵), 𝑦 > Next, we have only to show that 𝑥 = 𝑥1 + 𝑥𝑠 > for all 𝑡 > 0; that is, if 𝜆 is small, then 𝑥 is near 𝑥𝑠 in the sup norm of 𝐵 Since 𝑥𝑠 is bounded away from zero, so is 𝑥 According to Theorem 5, for system (2), 𝑥 is near 𝑥𝑠 , which implies that 𝑧 is near 𝑧𝑠 We notice that the period of the periodic solution (𝑥, 𝑦) is 𝜏; therefore, 𝑧 = 𝑠̃ − 𝑥 − 𝑦 > is piecewise continuous and 𝜏-periodic The proof is completed Numerical Simulations In order to justify our theoretic results, we will give some numerical simulations Let the parameters of system (2) be 𝑎1 = 0.2, 𝑎2 = 0.2, 𝑚1 = 1.3, 𝑚2 = 1.3, 𝑞 = 0.16, and 𝑇 = 0.8 By computing, 𝑚2∗ = 1.404 > 𝑚2 = 1.3 is obtained From Theorem 5, we know that if 𝑚2∗ > 𝑚2 , system (2) has a unique globally asymptotically stable boundary 𝜏-periodic solution (̃𝑠(𝑡), 0, 0), which is shown in Figure Suppose that 𝑎1 = 0.2, 𝑎2 = 0.2, 𝑚1 = 1.3, 𝑚2 = 1.5, 𝑞 = 0.2, and 𝑇 = 0.8 By computing, we have 𝑚2∗ = 1.410 < 𝑚2 = 1.5, 𝑚1∗ = 1.763 > 𝑚1 = 1.3 According to Theorem 5, for 𝑚2 > 𝑚2∗ and 𝑚1 < 𝑚1∗ , system (2) has a unique globally asymptotically stable boundary 𝜏-periodic solution (𝑥𝑠 (𝑡)), 0, 𝑧𝑠 (𝑡), which is demonstrated in Figure Set 𝑎1 = 0.2, 𝑎2 = 0.2, 𝑚1 = 2, 𝑚2 = 1.5, 𝑞 = 0.2, and 𝑇 = 0.8 By computing, we obtain 𝑚1∗ = 1.763 < 𝑚1 = 2, 𝑚2∗ = 1.410 < 𝑚2 = 1.5 From Theorem 5, when 𝑚2 > 𝑚2∗ and 𝑚1 > 𝑚1∗ , system (2) exists a positive periodic solution, which is simulated in Figure Journal of Applied Mathematics Discussion In this paper, we have investigated a competitive model of plasmid-bearing and plasmid-free organisms in a pulsed chemostat It is important in biotechnology for the study of plasmid stability where the effects of plasmid loss in genetically altered organisms (the plasmid-free organism is presumably the better competitor) are investigated Firstly, we find the invasion threshold of the plasmid-free organism, 𝜏 which is 𝑚2∗ = (𝜏/ ∫0 (̃𝑠(𝑙)/(𝑎1 + 𝑠̃(𝑙) + 𝑎2 𝑠̃2 (𝑙)))𝑑𝑙) If 𝑚2 < 𝑚2∗ , the microorganism-free periodic solution (̃𝑠(0), 0, 0) is globally asymptotically stable, which is simulated in Figure If 𝑚2 > 𝑚2∗ , then the plasmid-free organism begins to invade the system Furthermore, we have proved that if 𝑚2 > 𝑚2∗ , 𝜏 there exists 𝑚1∗ = (𝜏/ ∫0 ((1−𝑞)𝑥𝑠 (𝑙)/(𝑎1 +𝑥𝑠 (𝑙)+𝑎2 𝑥𝑠2 (𝑙)))𝑑𝑙) as the invasion threshold of the plasmid-bearing; that is to say, if 𝑚1 < 𝑚1∗ the boundary periodic solution (𝑥𝑠 (𝑡), 0, 𝑧𝑠 (𝑡)) is globally asymptotically stable (see Figure 2) and if 𝑚1 > 𝑚1∗ , the solution (𝑥𝑠 (𝑡), 0, 𝑧𝑠 (𝑡)) is unstable Lastly by using standard techniques of bifurcation theory, we prove that if 𝑚2 > 𝑚2∗ and 𝑎1 − 𝑎2 𝑥𝑠2 ≥ 0, the system has a positive 𝜏-periodic solution, which is shown in Figure System (2) compared with other literatures is more interesting since it can be used to model the excess substrate inhibition in the process of the genetically modified fermentation We show that the plasmid-bearing and plasmid-free can coexist on a periodic solution The analysis indicates that the change of one organism into another can complicate the dynamics of system (2) since the plasmid-bearing concentration is relevant to the substrate concentration and the substrate concentration can be controlled by setting the period of impulsive input However, how to find the optimal period of impulsive input makes microorganisms reach the maximum production by genetically altered biotechnology, which is a challenging problem to solve and we will leave it for the future Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper Acknowledgments This work is supported by the National Natural Science Foundation of China (no 11371164), NSFC-Talent Training Fund of Henan (no U1304104), the young backbone teachers of Henan (no 2013GGJS-214), and Henan Science and Technology Department (nos 132300410084 and 132300410250) References [1] F Mazenc, M Malisoff, and P De Leenheer, “On the stability of periodic solutions in the perturbed chemostat,” Mathematical Biosciences and Engineering, vol 4, no 2, pp 319–338, 2007 [2] F Mazenc, M Malisoff, and J Harmand, “Stabilization and robustness analysis for a chemostat model with two species and monod growth rates via a Lyapunov approach,” in Proceedings [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] of the 46th IEEE Conference on Decision and Control, pp 3933– 3938, New Orleans, La, USA, December 2007 F Mazenc, M Malisoff, and J Harmand, “Further results on stabilization of periodic trajectories for a chemostat with two species,” IEEE Transactions on Automatic Control, vol 53, pp 66–74, 2008 H Nie and J H Wu, “Positive solutions of a competition model for two resources in the unstirred chemostat,” Journal of Mathematical Analysis and Applications, vol 355, no 1, pp 231– 242, 2009 S L Yuan, W G Zhang, and M A Han, “Global asymptotic behavior in chemostat-type competition models with delay,” Nonlinear Analysis Real World Applications, vol 10, no 3, pp 1305–1320, 2009 O Tagashira, “Permanent coexistence in chemostat models with delayed feedback control,” Nonlinear Analysis Real World Applications, vol 10, no 3, pp 1443–1452, 2009 M I Nelson and H S Sidhu, “Analysis of a chemostat model with variable yield coefficient,” Journal of Mathematical Chemistry, vol 38, no 4, pp 605–615, 2005 M I Nelson and H S Sidhu, “Analysis of a chemostat model with variable yield coefficient: tessier kinetics,” Journal of Mathematical Chemistry, vol 46, no 2, pp 303–321, 2009 H L Smith and P Waltman, The Theory of the Chemostat, vol 13 of Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge, UK, 1995 Z Y Xiang and X Y Song, “A model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with periodic input,” Chaos, Solitons & Fractals, vol 32, no 4, pp 1419–1428, 2007 X Shi, X Song, and X Zhou, “Analysis of a model of plasmidbearing, plasmid-free competition in a pulsed chemostat,” Advances in Complex Systems, vol 9, no 3, pp 263–276, 2006 R Pal, D Basu, and M Banerjee, “Modelling of phytoplankton allelopathy with Monod-Haldane-type functional response— a mathematical study,” BioSystems, vol 95, no 3, pp 243–253, 2009 J M Cushing, “Periodic time-dependent predator-prey systems,” SIAM Journal on Applied Mathematics, vol 32, no 1, pp 82–95, 1977 Copyright of Journal of Applied Mathematics is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission However, users may print, download, or email articles for individual use  ... Song, and X Zhou, ? ?Analysis of a model of plasmidbearing, plasmid- free competition in a pulsed chemostat, ” Advances in Complex Systems, vol 9, no 3, pp 263–276, 2006 R Pal, D Basu, and M Banerjee,... Press, Cambridge, UK, 1995 Z Y Xiang and X Y Song, ? ?A model of competition between plasmid- bearing and plasmid- free organisms in a chemostat with periodic input, ” Chaos, Solitons & Fractals, vol... ? ?Analysis of a chemostat model with variable yield coefficient,” Journal of Mathematical Chemistry, vol 38, no 4, pp 605–615, 2005 M I Nelson and H S Sidhu, ? ?Analysis of a chemostat model with variable